libstdc++
modified_bessel_func.tcc
1 // Special functions -*- C++ -*-
2 
3 // Copyright (C) 2006, 2007, 2008, 2009
4 // Free Software Foundation, Inc.
5 //
6 // This file is part of the GNU ISO C++ Library. This library is free
7 // software; you can redistribute it and/or modify it under the
8 // terms of the GNU General Public License as published by the
9 // Free Software Foundation; either version 3, or (at your option)
10 // any later version.
11 //
12 // This library is distributed in the hope that it will be useful,
13 // but WITHOUT ANY WARRANTY; without even the implied warranty of
14 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 // GNU General Public License for more details.
16 //
17 // Under Section 7 of GPL version 3, you are granted additional
18 // permissions described in the GCC Runtime Library Exception, version
19 // 3.1, as published by the Free Software Foundation.
20 
21 // You should have received a copy of the GNU General Public License and
22 // a copy of the GCC Runtime Library Exception along with this program;
23 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
24 // <http://www.gnu.org/licenses/>.
25 
26 /** @file tr1/modified_bessel_func.tcc
27  * This is an internal header file, included by other library headers.
28  * You should not attempt to use it directly.
29  */
30 
31 //
32 // ISO C++ 14882 TR1: 5.2 Special functions
33 //
34 
35 // Written by Edward Smith-Rowland.
36 //
37 // References:
38 // (1) Handbook of Mathematical Functions,
39 // Ed. Milton Abramowitz and Irene A. Stegun,
40 // Dover Publications,
41 // Section 9, pp. 355-434, Section 10 pp. 435-478
42 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
43 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
44 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
45 // 2nd ed, pp. 246-249.
46 
47 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
48 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
49 
50 #include "special_function_util.h"
51 
52 namespace std
53 {
54 namespace tr1
55 {
56 
57  // [5.2] Special functions
58 
59  // Implementation-space details.
60  namespace __detail
61  {
62 
63  /**
64  * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and
65  * @f$ K_\nu(x) @f$ and their first derivatives
66  * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
67  * These four functions are computed together for numerical
68  * stability.
69  *
70  * @param __nu The order of the Bessel functions.
71  * @param __x The argument of the Bessel functions.
72  * @param __Inu The output regular modified Bessel function.
73  * @param __Knu The output irregular modified Bessel function.
74  * @param __Ipnu The output derivative of the regular
75  * modified Bessel function.
76  * @param __Kpnu The output derivative of the irregular
77  * modified Bessel function.
78  */
79  template <typename _Tp>
80  void
81  __bessel_ik(const _Tp __nu, const _Tp __x,
82  _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
83  {
84  if (__x == _Tp(0))
85  {
86  if (__nu == _Tp(0))
87  {
88  __Inu = _Tp(1);
89  __Ipnu = _Tp(0);
90  }
91  else if (__nu == _Tp(1))
92  {
93  __Inu = _Tp(0);
94  __Ipnu = _Tp(0.5L);
95  }
96  else
97  {
98  __Inu = _Tp(0);
99  __Ipnu = _Tp(0);
100  }
101  __Knu = std::numeric_limits<_Tp>::infinity();
102  __Kpnu = -std::numeric_limits<_Tp>::infinity();
103  return;
104  }
105 
106  const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
107  const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
108  const int __max_iter = 15000;
109  const _Tp __x_min = _Tp(2);
110 
111  const int __nl = static_cast<int>(__nu + _Tp(0.5L));
112 
113  const _Tp __mu = __nu - __nl;
114  const _Tp __mu2 = __mu * __mu;
115  const _Tp __xi = _Tp(1) / __x;
116  const _Tp __xi2 = _Tp(2) * __xi;
117  _Tp __h = __nu * __xi;
118  if ( __h < __fp_min )
119  __h = __fp_min;
120  _Tp __b = __xi2 * __nu;
121  _Tp __d = _Tp(0);
122  _Tp __c = __h;
123  int __i;
124  for ( __i = 1; __i <= __max_iter; ++__i )
125  {
126  __b += __xi2;
127  __d = _Tp(1) / (__b + __d);
128  __c = __b + _Tp(1) / __c;
129  const _Tp __del = __c * __d;
130  __h *= __del;
131  if (std::abs(__del - _Tp(1)) < __eps)
132  break;
133  }
134  if (__i > __max_iter)
135  std::__throw_runtime_error(__N("Argument x too large "
136  "in __bessel_jn; "
137  "try asymptotic expansion."));
138  _Tp __Inul = __fp_min;
139  _Tp __Ipnul = __h * __Inul;
140  _Tp __Inul1 = __Inul;
141  _Tp __Ipnu1 = __Ipnul;
142  _Tp __fact = __nu * __xi;
143  for (int __l = __nl; __l >= 1; --__l)
144  {
145  const _Tp __Inutemp = __fact * __Inul + __Ipnul;
146  __fact -= __xi;
147  __Ipnul = __fact * __Inutemp + __Inul;
148  __Inul = __Inutemp;
149  }
150  _Tp __f = __Ipnul / __Inul;
151  _Tp __Kmu, __Knu1;
152  if (__x < __x_min)
153  {
154  const _Tp __x2 = __x / _Tp(2);
155  const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
156  const _Tp __fact = (std::abs(__pimu) < __eps
157  ? _Tp(1) : __pimu / std::sin(__pimu));
158  _Tp __d = -std::log(__x2);
159  _Tp __e = __mu * __d;
160  const _Tp __fact2 = (std::abs(__e) < __eps
161  ? _Tp(1) : std::sinh(__e) / __e);
162  _Tp __gam1, __gam2, __gampl, __gammi;
163  __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
164  _Tp __ff = __fact
165  * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
166  _Tp __sum = __ff;
167  __e = std::exp(__e);
168  _Tp __p = __e / (_Tp(2) * __gampl);
169  _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
170  _Tp __c = _Tp(1);
171  __d = __x2 * __x2;
172  _Tp __sum1 = __p;
173  int __i;
174  for (__i = 1; __i <= __max_iter; ++__i)
175  {
176  __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
177  __c *= __d / __i;
178  __p /= __i - __mu;
179  __q /= __i + __mu;
180  const _Tp __del = __c * __ff;
181  __sum += __del;
182  const _Tp __del1 = __c * (__p - __i * __ff);
183  __sum1 += __del1;
184  if (std::abs(__del) < __eps * std::abs(__sum))
185  break;
186  }
187  if (__i > __max_iter)
188  std::__throw_runtime_error(__N("Bessel k series failed to converge "
189  "in __bessel_jn."));
190  __Kmu = __sum;
191  __Knu1 = __sum1 * __xi2;
192  }
193  else
194  {
195  _Tp __b = _Tp(2) * (_Tp(1) + __x);
196  _Tp __d = _Tp(1) / __b;
197  _Tp __delh = __d;
198  _Tp __h = __delh;
199  _Tp __q1 = _Tp(0);
200  _Tp __q2 = _Tp(1);
201  _Tp __a1 = _Tp(0.25L) - __mu2;
202  _Tp __q = __c = __a1;
203  _Tp __a = -__a1;
204  _Tp __s = _Tp(1) + __q * __delh;
205  int __i;
206  for (__i = 2; __i <= __max_iter; ++__i)
207  {
208  __a -= 2 * (__i - 1);
209  __c = -__a * __c / __i;
210  const _Tp __qnew = (__q1 - __b * __q2) / __a;
211  __q1 = __q2;
212  __q2 = __qnew;
213  __q += __c * __qnew;
214  __b += _Tp(2);
215  __d = _Tp(1) / (__b + __a * __d);
216  __delh = (__b * __d - _Tp(1)) * __delh;
217  __h += __delh;
218  const _Tp __dels = __q * __delh;
219  __s += __dels;
220  if ( std::abs(__dels / __s) < __eps )
221  break;
222  }
223  if (__i > __max_iter)
224  std::__throw_runtime_error(__N("Steed's method failed "
225  "in __bessel_jn."));
226  __h = __a1 * __h;
227  __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
228  * std::exp(-__x) / __s;
229  __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
230  }
231 
232  _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
233  _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
234  __Inu = __Inumu * __Inul1 / __Inul;
235  __Ipnu = __Inumu * __Ipnu1 / __Inul;
236  for ( __i = 1; __i <= __nl; ++__i )
237  {
238  const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
239  __Kmu = __Knu1;
240  __Knu1 = __Knutemp;
241  }
242  __Knu = __Kmu;
243  __Kpnu = __nu * __xi * __Kmu - __Knu1;
244 
245  return;
246  }
247 
248 
249  /**
250  * @brief Return the regular modified Bessel function of order
251  * \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
252  *
253  * The regular modified cylindrical Bessel function is:
254  * @f[
255  * I_{\nu}(x) = \sum_{k=0}^{\infty}
256  * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
257  * @f]
258  *
259  * @param __nu The order of the regular modified Bessel function.
260  * @param __x The argument of the regular modified Bessel function.
261  * @return The output regular modified Bessel function.
262  */
263  template<typename _Tp>
264  _Tp
265  __cyl_bessel_i(const _Tp __nu, const _Tp __x)
266  {
267  if (__nu < _Tp(0) || __x < _Tp(0))
268  std::__throw_domain_error(__N("Bad argument "
269  "in __cyl_bessel_i."));
270  else if (__isnan(__nu) || __isnan(__x))
271  return std::numeric_limits<_Tp>::quiet_NaN();
272  else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
273  return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
274  else
275  {
276  _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
277  __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
278  return __I_nu;
279  }
280  }
281 
282 
283  /**
284  * @brief Return the irregular modified Bessel function
285  * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
286  *
287  * The irregular modified Bessel function is defined by:
288  * @f[
289  * K_{\nu}(x) = \frac{\pi}{2}
290  * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
291  * @f]
292  * where for integral \f$ \nu = n \f$ a limit is taken:
293  * \f$ lim_{\nu \to n} \f$.
294  *
295  * @param __nu The order of the irregular modified Bessel function.
296  * @param __x The argument of the irregular modified Bessel function.
297  * @return The output irregular modified Bessel function.
298  */
299  template<typename _Tp>
300  _Tp
301  __cyl_bessel_k(const _Tp __nu, const _Tp __x)
302  {
303  if (__nu < _Tp(0) || __x < _Tp(0))
304  std::__throw_domain_error(__N("Bad argument "
305  "in __cyl_bessel_k."));
306  else if (__isnan(__nu) || __isnan(__x))
307  return std::numeric_limits<_Tp>::quiet_NaN();
308  else
309  {
310  _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
311  __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
312  return __K_nu;
313  }
314  }
315 
316 
317  /**
318  * @brief Compute the spherical modified Bessel functions
319  * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
320  * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
321  * respectively.
322  *
323  * @param __n The order of the modified spherical Bessel function.
324  * @param __x The argument of the modified spherical Bessel function.
325  * @param __i_n The output regular modified spherical Bessel function.
326  * @param __k_n The output irregular modified spherical
327  * Bessel function.
328  * @param __ip_n The output derivative of the regular modified
329  * spherical Bessel function.
330  * @param __kp_n The output derivative of the irregular modified
331  * spherical Bessel function.
332  */
333  template <typename _Tp>
334  void
335  __sph_bessel_ik(const unsigned int __n, const _Tp __x,
336  _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
337  {
338  const _Tp __nu = _Tp(__n) + _Tp(0.5L);
339 
340  _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
341  __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
342 
343  const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
344  / std::sqrt(__x);
345 
346  __i_n = __factor * __I_nu;
347  __k_n = __factor * __K_nu;
348  __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
349  __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
350 
351  return;
352  }
353 
354 
355  /**
356  * @brief Compute the Airy functions
357  * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
358  * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
359  * respectively.
360  *
361  * @param __n The order of the Airy functions.
362  * @param __x The argument of the Airy functions.
363  * @param __i_n The output Airy function.
364  * @param __k_n The output Airy function.
365  * @param __ip_n The output derivative of the Airy function.
366  * @param __kp_n The output derivative of the Airy function.
367  */
368  template <typename _Tp>
369  void
370  __airy(const _Tp __x,
371  _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
372  {
373  const _Tp __absx = std::abs(__x);
374  const _Tp __rootx = std::sqrt(__absx);
375  const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
376 
377  if (__isnan(__x))
378  return std::numeric_limits<_Tp>::quiet_NaN();
379  else if (__x > _Tp(0))
380  {
381  _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
382 
383  __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
384  __Ai = __rootx * __K_nu
385  / (__numeric_constants<_Tp>::__sqrt3()
386  * __numeric_constants<_Tp>::__pi());
387  __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
388  + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
389 
390  __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
391  __Aip = -__x * __K_nu
392  / (__numeric_constants<_Tp>::__sqrt3()
393  * __numeric_constants<_Tp>::__pi());
394  __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
395  + _Tp(2) * __I_nu
396  / __numeric_constants<_Tp>::__sqrt3());
397  }
398  else if (__x < _Tp(0))
399  {
400  _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
401 
402  __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
403  __Ai = __rootx * (__J_nu
404  - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
405  __Bi = -__rootx * (__N_nu
406  + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
407 
408  __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
409  __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
410  + __J_nu) / _Tp(2);
411  __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
412  - __N_nu) / _Tp(2);
413  }
414  else
415  {
416  // Reference:
417  // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
418  // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
419  __Ai = _Tp(0.35502805388781723926L);
420  __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
421 
422  // Reference:
423  // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
424  // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
425  __Aip = -_Tp(0.25881940379280679840L);
426  __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
427  }
428 
429  return;
430  }
431 
432  } // namespace std::tr1::__detail
433 }
434 }
435 
436 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC